It looks like the setting is getting set in a local sqlite database in storage:
storage/default/https+++assistant.kagi.com^userContextId=1/ls/data.sqlite
running strings on the file I get the followng dump:
)tabledatadata
CREATE TABLE data( key TEXT PRIMARY KEY, utf16_length INTEGER NOT NULL, conversion_type INTEGER NOT NULL, compression_type INTEGER NOT NULL, last_access_time INTEGER NOT NULL DEFAULT 0, value BLOB NOT NULL)'
indexsqlite_autoindex_data_1data
9tabledatabasedatabase
CREATE TABLE database( origin TEXT NOT NULL, usage INTEGER NOT NULL DEFAULT 0, last_vacuum_time INTEGER NOT NULL DEFAULT 0, last_analyze_time INTEGER NOT NULL DEFAULT 0, last_vacuum_size INTEGER NOT NULL DEFAULT 0)
https://assistant.kagi.com^userContextId=1@
ui:theme
dark&
folders:section-collapsed
false
selected_model
ki_quick
ui:theme
folders:section-collapsed
) selected_model